+179 Let \(S\) be the set of nonzero real numbers. Let \(f : S \to \mathbb{R}\) be a function such that
(i) \(f(1) = 1\)
(ii) \(f \left( \frac{1}{x + y} \right) = f \left( \frac{1}{x} \right) + f \left( \frac{1}{y} \right)\) for all \(x, y \in S\) such that \(x + y \in S\) and
(iii)\((x + y) f(x + y) = xyf(x)f(y)\)for all \(x, y \in S\) such that \(x + y \in S\)
Find the number of possible functions \(f(x)\).
From experimenting and algebra, \(f(x)=\frac{1}{x}\) is a valid solution. I just want to know if there are any more. If I get the question right, the website will show me an explanation of the solution, so feel free to leave out complex LaTeX formatting if you want.
